A Discrete Carleson Theorem Along the Primes with a Restricted Supremum

Abstract

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2\pi i \lambda p}}{p} |,\] where $\pm \mathbb{P} := \{ \pm p : p \text{ is a prime} \}$, and $\Lambda \subset [0,1]$. We give sufficient conditions on $\Lambda$, met by (finite unions of) lacunary sets, for this to be a bounded sublinear operator on $\ell^p(\mathbb{Z})$ for $\frac{3}{2} < p < 4$.